Happy birthday to no one, not today. A birthday problem

According to Facebook, none of my friends have a birthday today. How many friends do I have to have so that there's a 50/50 chance that there will be a birthday every day of the year?

An answer comes from the blog Math Goes Pop!, referencing the well known "coupon collector's problem".

What’s known about this problem? Well, as I said above, even if you buy hundreds of thousands of cards, or stuff millions of people in a room, there’s no guarantee that you’ll collect every card or every date. However, on average, the number of cards you’ll need to go through to complete a set of size n is about n*log n.

In terms of birthdays, this says that if you want to collect every date, on average you’ll need to pool together around 2,153 people. Why such a large number? It’s not unreasonable to expect something like this – when you first begin collecting people, it won’t be hard to get people with different birthdays. However, as your numbers increase, you’ll get a new birthday less and less frequently. Finding that last birthday could prove to be quite elusive.

No birthdays today means no existential angst about filling in witty, brief birthday greetings to all of your friends. You could have swipe file of pithy and varied greetings that can be cut and pasted in with loving care. (Especially if you have 2,153 people to greet annually.)


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